This cannot be bounded via conventional means, i.e. using properties of the CDF, due to the oscillatory nature of the cosine function in the exponent. Bounding it directly using the Lipschitz continuity of \(h\)  will result in very large upper bounds.
In addition to this, one usually relies upon applying one or more of: exchangeable pairs, size-biasing, sum of variables and zero-biasing in order to bound the Wasserstein metric. This will not be needed when working with the kernel as our method — reminiscent of [1]} — shall be directly comparing the operators to obtain an upper bound. As we shall see, there are limitations to which distributions we can compare, which is the price to pay for using this kernel based method.
